3.5.48 \(\int \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{3/2} \, dx\) [448]

Optimal. Leaf size=219 \[ \frac {3 \sqrt {c-a^2 c x^2} \sqrt {\text {ArcSin}(a x)}}{16 a \sqrt {1-a^2 x^2}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\text {ArcSin}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}} \]

[Out]

1/2*x*arcsin(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)+1/5*arcsin(a*x)^(5/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-3
/32*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)+3/16*(-a^2*c*x^2
+c)^(1/2)*arcsin(a*x)^(1/2)/a/(-a^2*x^2+1)^(1/2)-3/8*a*x^2*(-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(1/2)/(-a^2*x^2+1)
^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4741, 4737, 4725, 4809, 3393, 3385, 3433} \begin {gather*} -\frac {3 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {\text {ArcSin}(a x)^{5/2} \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \text {ArcSin}(a x)^{3/2} \sqrt {c-a^2 c x^2}-\frac {3 a x^2 \sqrt {\text {ArcSin}(a x)} \sqrt {c-a^2 c x^2}}{8 \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {\text {ArcSin}(a x)} \sqrt {c-a^2 c x^2}}{16 a \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2),x]

[Out]

(3*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(16*a*Sqrt[1 - a^2*x^2]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[
a*x]])/(8*Sqrt[1 - a^2*x^2]) + (x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/2 + (Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^
(5/2))/(5*a*Sqrt[1 - a^2*x^2]) - (3*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(32
*a*Sqrt[1 - a^2*x^2])

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[f*(x^2/d)],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4725

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcSin[c*x])^n/(m
+ 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4741

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*((
a + b*ArcSin[c*x])^n/2), x] + (Dist[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[x*(a + b*ArcSin[c*x])^(
n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4809

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*c
^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x],
x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m,
0]

Rubi steps

\begin {align*} \int \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2} \, dx &=\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\sin ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{2 \sqrt {1-a^2 x^2}}-\frac {\left (3 a \sqrt {c-a^2 c x^2}\right ) \int x \sqrt {\sin ^{-1}(a x)} \, dx}{4 \sqrt {1-a^2 x^2}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}+\frac {\left (3 a^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2}{\sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin ^2(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a \sqrt {1-a^2 x^2}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{16 a \sqrt {1-a^2 x^2}}\\ &=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{16 a \sqrt {1-a^2 x^2}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}-\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a \sqrt {1-a^2 x^2}}\\ &=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{16 a \sqrt {1-a^2 x^2}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}-\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}\\ &=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{16 a \sqrt {1-a^2 x^2}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {\pi } \sqrt {c-a^2 c x^2} C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.08, size = 158, normalized size = 0.72 \begin {gather*} \frac {\sqrt {c-a^2 c x^2} \sqrt {\text {ArcSin}(a x)} \left (32 \text {ArcSin}(a x) \sqrt {\text {ArcSin}(a x)^2} \left (5 a x \sqrt {1-a^2 x^2}+2 \text {ArcSin}(a x)\right )+15 \sqrt {2} \sqrt {i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {3}{2},-2 i \text {ArcSin}(a x)\right )+15 \sqrt {2} \sqrt {-i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {3}{2},2 i \text {ArcSin}(a x)\right )\right )}{320 a \sqrt {1-a^2 x^2} \sqrt {\text {ArcSin}(a x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2),x]

[Out]

(Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]]*(32*ArcSin[a*x]*Sqrt[ArcSin[a*x]^2]*(5*a*x*Sqrt[1 - a^2*x^2] + 2*ArcSin
[a*x]) + 15*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (-2*I)*ArcSin[a*x]] + 15*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gam
ma[3/2, (2*I)*ArcSin[a*x]]))/(320*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]^2])

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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \sqrt {-a^{2} c \,x^{2}+c}\, \arcsin \left (a x \right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),x)

[Out]

int((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**(1/2)*asin(a*x)**(3/2),x)

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*asin(a*x)**(3/2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {asin}\left (a\,x\right )}^{3/2}\,\sqrt {c-a^2\,c\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^(3/2)*(c - a^2*c*x^2)^(1/2),x)

[Out]

int(asin(a*x)^(3/2)*(c - a^2*c*x^2)^(1/2), x)

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